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<p>A differential equation that has <dfn class="terminology">unknown functions</dfn> and <dfn class="terminology">their partial derivatives</dfn>.</p>
<p><span class="process-math">\(\circ\)</span>  <span class="process-math">\(\displaystyle u_{tt}=a^2u_{xx}\text{,}\)</span>  1D wave eqn<span class="process-math">\(\circ\)</span>  <span class="process-math">\(\displaystyle\left(\frac{\partial^2u}{\partial x^2}\right)^2+\frac{\partial^2 u}{\partial x\partial y}=x^2+y^2\text{.}\)</span><span class="process-math">\(\circ\)</span>  <span class="process-math">\(\displaystyle\frac{\partial^2u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0\text{,}\)</span> 2D Laplace eqn<span class="process-math">\(\circ\)</span>  <span class="process-math">\(\displaystyle\frac{\partial^2u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=f(x,y)\text{,}\)</span>  2D Poisson eqn—————————————————————————————————————————- –</p>
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<li><p>: The highest order of derivatives.</p></li>
<li><p>: The terms with <span class="process-math">\(u\)</span> and its derivatives are in a <dfn class="terminology">linear</dfn> form. Otherwise, <dfn class="terminology">nonlinear</dfn>.</p></li>
<li><p>: If all the terms contains <span class="process-math">\(u\)</span> and its derivatives, then it is <dfn class="terminology">homogeneous</dfn>, otherwise <dfn class="terminology">non-homogeneous</dfn>.</p></li>
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